A Radio Frequency Fingerprint Recognition Method

Abstract

The present invention relates to a radio frequency fingerprint recognition method, belonging to the technical fields of machine learning, physical layer security, and signal classification. The radio frequency fingerprint recognition method transforms the received signal into a constellation trajectory diagram, then clusters the constellation trajectory diagram to obtain the clustering center points, calculates the within-class average distance and the sum of within-class distances of each cluster, constructs the within-class average distance and the sum of within-class distances of the corresponding cluster as a feature vector, which serves as the radio frequency fingerprint feature of the device, and then conducts classification, that is, identifies the device according to the features. The method has low time complexity and space complexity, and has a high recognition accuracy under the condition of small sample data; compared with transient signals, the radio frequency fingerprints of steady-state signals contain more hardware information of wireless devices, so they are easier to extract and classify and recognize; compared with the shape image recognition of signal distribution, the preprocessing process is relatively simple.

Description

Technical Field

[0001] The present invention relates to a radio frequency fingerprint recognition method, belonging to the technical fields of machine learning, physical layer security, and signal classification. Background Art

[0002] Due to the openness of wireless networks, the risk of illegal user access and large-scale malicious attacks has increased, which has become one of the factors seriously hindering the development and application of wireless network technologies. "Radio frequency fingerprint" is the unique feature of a device extracted by analyzing its wireless signal, which is the essential feature of the physical layer of the device and is only related to the hardware features of the device itself. Using radio frequency fingerprints to identify the identities of different wireless devices can effectively detect wireless devices with disguised identities, thereby improving the security of wireless networks. By transforming the received signal into a constellation trajectory diagram to obtain its statistical features, radio frequency fingerprint features of the device can be extracted without prior knowledge of the device's transmitted signal, which is applicable to physical layer security and the identification and authentication of wireless access devices. Since the carrier frequency offsets of different wireless devices are different, the phase rotations generated by their constellation trajectory diagrams are also different. Therefore, by examining the similarity of the phase rotations in the differential constellation trajectory diagram, it can be determined whether different signals come from the same wireless device, which is used as the basis for wireless device identity recognition.

[0003] Support vector machines (SVM) is a binary classification model, including one-class support vector machines and two-class support vector machines. The traditional SVM model is a linear classifier and is the maximum margin linear classifier defined in the feature space. The learning strategy of SVM is margin maximization, which essentially is a convex quadratic programming problem and is also equivalent to the problem of minimizing the regularized hinge loss function. Compared with two-class support vector machines, SV-means belongs to a one-class support vector machine model. This model only requires one type of data during training and regards the origin as another class of data different from the provided training set. This model does not require providing data with positive labels (PL) and data with negative labels (NL) simultaneously during training. Compared with two-class support vector machines, one-class support vector machines are more suitable for practical problems such as anomaly detection and single training sample recognition.

[0004] The objective of the present invention is to address the technical drawbacks of high time complexity and algorithm complexity in the above algorithms while ensuring a high classification accuracy. In terms of classification accuracy, the SV-means method is comparable to SVM, BPNN, and CNN; however, in terms of complexity, the advantages of SV-means are prominent: this method has an extremely low computational complexity, which in turn leads to an extremely low algorithm time delay, making this method more suitable for the classification of small samples. Summary of the Invention

[0005] The objective of the present invention is to propose a radio frequency fingerprint recognition method for the existing radio frequency fingerprint recognition algorithms with low accuracy and low complexity. By transforming the received signal into a constellation trajectory diagram, clustering the constellation trajectory diagram to obtain the clustering center points, calculating the within-class average distance and the sum of within-class distances of each cluster, constructing the within-class average distance and the sum of within-class distances of the corresponding clusters into a feature vector as the radio frequency fingerprint feature of the device, and then performing classification, that is, identifying the device according to the feature.

[0006] To achieve the above objective, the present invention adopts the following technical solutions.

[0007] The radio frequency fingerprint recognition method relies on a system including M transmitting devices and a receiver; N samples are collected for each segment of the signal; the signal-to-noise ratio range is 5 to 25 dB;

[0008] Step 1: M different devices transmit modulated signals, and after the receiver receives the signals, it processes them to obtain a constellation trajectory diagram, which specifically includes the following sub-steps:

[0009] Step 1.1: N samples are collected for each segment of the signal, sampled at a sampling rate that satisfies the Nyquist law, and the energy of the sampled signal is normalized;

[0010] Step 1.2: The I-channel and Q-channel signals in the energy-normalized signal output in Step 1.1 are respectively controlled by a delay device for the delay of the I-channel and Q-channel, and differential processing is performed at a certain interval;

[0011] Step 1.3: The differentially processed signal output in Step 1.2 is directly plotted in the space with the I-channel and Q-channel as the coordinate axes to obtain the corresponding constellation trajectory diagram;

[0012] Step 2: Cluster the constellation points in the constellation trajectory diagram output in Step 1.3, calculate the sum of within-class distances and the within-class average distance vector, use the obtained clustering center points as the radio frequency fingerprint, and construct a feature vector set;

[0013] Step 2 is specifically as follows:

[0014] Step 2.1: Cluster the constellation points in the constellation trajectory diagram output in Step 1 to obtain different cluster centers of the densely distributed points in the constellation trajectory diagram;

[0015] Step 2.2: According to the different cluster centers of the densely distributed points in the constellation trajectory diagram obtained in Step 2.1, calculate the sum of the within-class distances and the within-class average distance vectors of each cluster;

[0016] Step 2.3: Sort the sum of the within-class distances and the within-class average distance vectors of each cluster calculated in a clockwise order according to the phase size of the cluster center points of each cluster, obtain the sorted sum of the within-class distances and the within-class average distance vectors of each cluster, and form a feature vector set;

[0017] Step 2.4: Traverse different signal-to-noise ratios and M devices, and repeat Steps 1 to 2.3 to obtain feature vector sets for signal extraction of different devices under different signal-to-noise ratios;

[0018] Step 3: Select A groups of feature vectors from the feature vector set as training samples for classification and B groups as test samples for classification, and respectively form a training set and a test set with the training samples and the test samples;

[0019] Step 4: Use the training set for training to obtain the trained classification model parameters;

[0020] Among them, the classification model parameters include a weight vector and a separation margin; the normal vector of the separation hyperplane in the weight vector feature space is denoted as w; the separation margin is the distance between the separation surface and the coordinate origin, denoted as ρ;

[0021] Step 4, the specific process includes the following sub-steps:

[0022] Step 4.1: Perform random Fourier feature transformation on the training samples in the training set to obtain a feature vector data set Z after random Fourier feature transformation;

[0023] Among them, the random Fourier feature transformation calculation formula is Equation (1):

[0024]

[0025] Among them, x i is the data of the training set, and the feature dimension is d, and the number of data is n; P(·) is the Gaussian distribution, σ is the kernel bandwidth, h u (d×1) is a randomly generated matrix; u = 1,..., d RF ; d RF is the dimension of the random Fourier feature transformation, z(x i ) is the corresponding item of x i in the feature set Z;

[0026] Step 4.2: Randomly select a data point from the feature vector dataset Z after random Fourier feature transformation as the initial weight vector w;

[0027] Step 4.3: Calculate the separation margin ρ based on w through golden-section line search;

[0028] Step 4.4: For all data points z(x j ) where wz(x j ) - ρ < 0, calculate the average to obtain the weight vector w m ;

[0029] where x j is the data in the training set, and z(x j ) is the corresponding item of x j in the feature vector dataset Z;

[0030] Step 4.5: Calculate a new weight vector w m between w and w new using the stochastic gradient descent method, and then update the weight vector w with w new ;

[0031] where updating the weight vector w with w new means setting w = w new ;

[0032] Step 4.6: Repeat steps 4.3 to 4.5 until w basically stabilizes after K iterations. Denote w as the optimal weight vector w * , and calculate the optimal separation margin ρ * corresponding to the optimal weight vector w * ;

[0033] where the value range of K is greater than 15 and less than 50, and the condition for basically stabilizing is that the modulus difference of w is less than 0.0001;

[0034] where the optimal weight vector w * and the optimal separation margin ρ * are the trained model parameters;

[0035] Step 5: Classify the test set, which specifically includes the following sub-steps:

[0036] Step 5.1: Perform random Fourier feature transformation on the test set, and obtain the test feature vector dataset Z according to formula (1) t ;

[0037] Step 5.2: Calculate w * , ρ * based on w * zt (x ti )-ρ * The result is denoted as P;

[0038] Among them, the dimension of P is 1×n t ; n t is the number of elements in the test set; x ti is the i-th element in the test set, and z t (x ti ) is an element in the test feature vector dataset Z ti after the random Fourier feature transformation of x t ;

[0039] Step 5.3: Make a judgment based on the element values of P obtained in Step 5.2. Specifically, if P(v)≥0, then the data to be classified corresponding to the v-th element in the test set is of the same type as the training set data; otherwise, if P(v)<0, then the data to be classified corresponding to the v-th element in the test set is of a different type from the training set data;

[0040] Among them, the value range of v is from 1 to n t .

[0041] Advantageous Effects

[0042] A radio frequency fingerprint recognition method according to the present invention has the following advantageous effects compared with the prior art:

[0043] 1. The method has a high accuracy rate for small samples;

[0044] 2. The time complexity and space complexity of the method are low;

[0045] 3. Compared with transient signals, the radio frequency fingerprints of steady-state signals contain more hardware information of wireless devices according to the method, so they are easier to extract and the classification and recognition performance is more stable;

[0046] 4. Compared with the shape image recognition of signal distribution, the preprocessing process of the method is relatively simple. Description of the Drawings

[0047] Figure 1 is a schematic flow chart of distinguishing between legal and illegal devices by using a radio frequency fingerprint recognition method according to the present invention;

[0048] Figure 2 is a schematic flow chart of distinguishing between multiple types of devices by using a radio frequency fingerprint recognition method according to the present invention. Detailed Embodiments

[0049] The following further illustrates and describes in detail a radio frequency fingerprint recognition method according to the present invention with reference to the drawings and embodiments.

[0050] Embodiment 1

[0051] This embodiment elaborates in detail the specific implementation of a radio frequency fingerprint recognition method of the present invention for classifying legal and illegal devices. Figure 1 It is a schematic diagram of a binary classification process.

[0052] Generally, the baseband steady-state response characteristics of wireless devices are used to characterize radio frequency fingerprints, mainly including carrier frequency offset, correlation value of synchronization signals, offset of baseband I / Q signals, amplitude and phase errors of demodulated signals, etc. Hardware defects of transmitter devices will cause modulation errors, and modulation errors will in turn affect the distribution of the constellation diagram. Use modulation errors or constellation diagrams to extract the radio frequency fingerprints of devices to distinguish different device individuals.

[0053] There are carrier frequency deviations between the transmitter and the receiver, and different devices have different frequency offsets; in actual operation, generally multiple transmitters have their own frequency offsets and phase differences, and the receiver is the same. The constellation diagram is obtained through differential processing, the deviation is amplified, and the devices are identified by extracting features.

[0054] During simulation, legal and illegal are to artificially set different phase offsets and frequency offsets at the receiving end to obtain different constellation diagrams and then store them. During testing, only one of the phase offsets and frequency offsets is used. If they match, it is legal; if they do not match, it is illegal. Legal and illegal are binary classifications.

[0055] When the method is specifically implemented, the steps are as follows:

[0056] Step 1: When M is 1, the device number to be recognized is recorded as 1, and a QPSK modulated signal is transmitted; when the receiver receives the signal and N is taken as 1, the signal constellation diagram data obtained by processing 1 segment of the signal specifically includes the following sub-steps:

[0057] Step 1.1: Sampling is performed at a sampling rate that satisfies the Nyquist law, and the energy of the sampled signal is normalized.

[0058] Step 1.2: The receiver receives the I / Q signals of the wireless device baseband; for the sampled signals of the I channel and the Q channel, a delay device is used to control the delays of the I channel and the Q channel respectively, and differential processing is performed at a certain interval. A stable constellation diagram is obtained through differentiation. The differential processing is shown in formula (2):

[0059]

[0060] In the formula, X(t) is the baseband signal of the transmitter, Y(t) is the baseband signal after frequency conversion, Y* is the conjugate value, and n is the differential interval.

[0061] Step 1.3: Directly plot the received signal in the space with the I-channel and Q-channel as the coordinate axes to obtain the corresponding constellation trajectory diagram;

[0062] Step 2: Obtain different clustering centers of the dense points in the constellation trajectory diagram through the k-means clustering algorithm, and use the obtained clustering center points as the RF fingerprints, specifically:

[0063] Step 2.1: Use k-means clustering to cluster the received signal to obtain the clustering result;

[0064] Step 2.2: According to the clustering result of Step 2.1, calculate the within-class distance sum and within-class average distance vector of each cluster;

[0065] Step 2.3: Arrange the within-class distance sum and within-class average distance vector of each calculated cluster in a clockwise order according to the phase size of the clustering center points of each cluster, obtain the sorted within-class distance sum and within-class average distance vector of each cluster, and form a feature vector set;

[0066] Step 3: According to the feature vector set extracted from the signal samples under different signal-to-noise ratios, select A groups of feature vectors as the training samples for classification and B groups as the test samples, and use SV-means for classification to obtain the classification accuracy. The specific process includes the following sub-steps:

[0067] In the specific implementation of Step 3.1, the random Fourier feature transform calculation formula is (3):

[0068]

[0069] where, z(x i ) represents the i-th data in the data set z, and this data is obtained by the random Fourier feature transform of x i through formula (3); x i has a dimension of 1×d, indicating that each data includes d types of feature, specifically in this embodiment, d = 100; the variation range of i is from 1 to n and n is the number of data, σ is the kernel bandwidth, and the value of σ in this instance is 0.5; h u is a randomly generated matrix, and the elements h u in the matrix follow the Gaussian distribution, and h u has a dimension of d×1, and the value range of the subscript u is from 1 to d RF / 2; d RF is the dimension of the random Fourier feature transform;

[0070] This step reduces the data dimension d RFArtificial control is carried out in 2000 dimensions, avoiding the problem of too high dimensions and uncontrollability caused by using kernel functions in traditional machine learning, greatly reducing the time complexity and space complexity of the algorithm, and achieving the effects of beneficial effects 1 and 2.

[0071] Step 3.2: Randomly select a data from the data set z as the initial weight vector w0;

[0072] Step 3.3: Calculate the separation distance ρ through the golden section line search;

[0073] Step 3.4: Cluster the data points where wz - ρ is less than 0, and then obtain the average weight vector after averaging;

[0074] Among them, the average weight vector is denoted as w m ;

[0075] Step 3.5: Select a new weight vector w between w and w through the stochastic gradient descent method m and assign w new to w; new

[0076] Step 3.6: Repeat steps 3.3 to 3.5 until after S iterations, obtain a stable w new ;

[0077] Step 4: Classify according to the calculated w and ρ; record the classification accuracy rates at different signal-to-noise ratios, and calculate the average classification accuracy rate at each specific signal-to-noise ratio. The results are shown in Table 1.

[0078] Table 1 Average classification accuracy rate at each specific signal-to-noise ratio under single-device conditions

[0079] Average recognition rate Kmeans+SVM The method described in this application Device 1 95.2％ 95.7％

[0080] Example 2

[0081] During simulation, for M multi-classifications, take N segments of data, and actually only take one segment of data during simulation. Manually set different phase offsets and frequency offsets at the receiving end to obtain different constellation diagrams and then store them. During testing, identify different devices by modifying the phase offset and frequency offset of the input signal several times;

[0082] The system on which the radio frequency fingerprint recognition method relies specifically includes the following steps:

[0083] Step 1: When M is 1, the device number to be recognized is denoted as 1, and a QPSK modulated signal is transmitted; the receiver receives the signal, N is taken as 1, and the signal constellation diagram data obtained by processing 1 segment of the signal specifically includes the following sub-steps:

[0084] Step 1.1: Sample at the sampling rate required to meet the Nyquist law and normalize the energy of the sampled signal;

[0085] Step 1.2: The receiver receives the I / Q signals of the wireless device baseband; for the sampled signals of the I channel and the Q channel, a delay device is used to control the delays of the I channel and the Q channel respectively, and differential processing is performed at a certain interval. A stable constellation diagram is obtained through differentiation. The differential processing is shown in Equation (2):

[0086]

[0087] In the formula, X(t) is the baseband signal of the transmitter, Y(t) is the baseband signal after frequency conversion, Y* is the conjugate value, and n is the interval of differentiation.

[0088] Step 1.3: Directly plot the received signal in the space with the I channel and the Q channel as the coordinate axes to obtain the corresponding constellation trajectory diagram;

[0089] Step 2: Use the Euclidean distance from each sample point to the origin as the feature vector set;

[0090] Step 3: According to the feature vector sets extracted from the signal samples under different signal-to-noise ratios, select A groups of feature vectors as the training samples for classification and B groups as the test samples, and use SV-means for classification to obtain the classification accuracy. The specific process includes the following sub-steps:

[0091] Step 3.1: Perform random Fourier feature transformation on the features to obtain the data set z after random Fourier feature transformation;

[0092] Step 3.2: Randomly select a data from the data set z as the initial weight vector w0;

[0093] Step 3.3: Calculate the separation distance ρ through the golden section line search;

[0094] Step 3.4: Cluster the data points where wz - ρ is less than 0, and then obtain the average weight vector after averaging; among them, the average weight vector is denoted as w m ;

[0095] Step 3.5: Select a new weight vector w m between w and w new through the stochastic gradient descent method, and assign w new to w;

[0096] Step 3.6: Repeat Step 3.3 to Step 3.5 until after S iterations, obtain a stable w new ;

[0097] Step 4: Classify according to the calculated \(w\) and \(\rho\); record the classification accuracy rates at different signal-to-noise ratios, and calculate the average classification accuracy rate at each specific signal-to-noise ratio.

[0098] Embodiment 3

[0099] The system on which the radio frequency fingerprint recognition method relies specifically includes the following steps:

[0100] Step 1: When \(M = 3\), the device numbers to be recognized are respectively denoted as 1, 2, and 3, and QPSK modulated signals are transmitted; after the receiver receives the signals, when \(N = 1\), the signal constellation map data obtained by processing 1 segment of the signals. After the receiver receives the signals, the obtained signal constellation map data specifically includes the following sub-steps:

[0101] Step 1.1: Sample at a sampling rate that satisfies the Nyquist law, and normalize the energy of the sampled signals.

[0102] Step 1.2: The receiving end receives the I / Q signals of the wireless device baseband; for the sampled signals of the I channel and the Q channel, a delay device is used to control the delays of the I channel and the Q channel respectively, and differential processing is performed at a certain interval to obtain a stable constellation map through differentiation. The differential processing is shown in formula (2):

[0103]

[0104] In the formula, \(X(t)\) is the baseband signal of the transmitter, \(Y(t)\) is the baseband signal after frequency conversion, \(Y^*\) is the conjugate value, and \(n\) is the differential interval.

[0105] Step 1.3: Directly plot the received signals in the space with the I channel and the Q channel as the coordinate axes to obtain the corresponding constellation trajectory diagram;

[0106] Step 2: Obtain different clustering centers of the dense points in the constellation trajectory diagram distribution through the k-means clustering algorithm, and use the obtained clustering center points as the radio frequency fingerprints, specifically:

[0107] Step 2.1: Use k-means clustering to cluster the received signals to obtain the clustering results;

[0108] Step 2.2: According to the clustering results in Step 2.1, calculate the Euclidean distances between the clustering center points of different devices;

[0109] Step 2.3: Then take the average value of the first \(K\) clustering center points as the reference clustering center point, and then calculate the Euclidean distances between the first \(L\) clustering center points and the reference clustering center point, and use this average value as the reference clustering center point;

[0110] Step 3: After obtaining the clustering center information of device f, for the newly input device f, calculate the Euclidean distance sum of its clustering centers as the feature vector; use SV-means for classification to obtain the classification accuracy. The specific process includes the following sub-steps:

[0111] Step 3.1: Perform random Fourier feature transformation on the features to obtain the data set z after random Fourier feature transformation;

[0112] This step artificially controls the data dimension d RF at 2000 dimensions, avoiding the problem of too high dimension and uncontrollability caused by using kernel functions in traditional machine learning, and greatly reducing the algorithm time complexity and space complexity.

[0113] Step 3.2: Randomly select a data from the data set z as the initial weight vector w0;

[0114] Step 3.3: Calculate the separation distance ρ through golden section line search;

[0115] Step 3.4: Cluster the data points where wz - ρ is less than 0, and then obtain the average weight vector after averaging;

[0116] Among them, the average weight vector is denoted as w m ;

[0117] Step 3.5: Select a new weight vector w m between w and w through the stochastic gradient descent method new , and assign w new to w;

[0118] Step 3.6: Repeat Step 3.3 to Step 3.5 until after S iterations, obtain a stable w new ;

[0119] Step 4: Classify according to the calculated w and ρ; record the classification accuracy under different signal-to-noise ratios, and calculate the average classification accuracy at each specific signal-to-noise ratio. The results are shown in Table 2, achieving beneficial effects 3 and 4.

[0120] Table 2 Average recognition rates of the present method and existing algorithms for 3 different devices

[0121]

[0122]

[0123] The above are the preferred embodiments of the present invention. The present invention should not be limited to the content disclosed in this embodiment and the drawings. Any equivalent or modified implementation completed without departing from the spirit disclosed by the present invention falls within the protection scope of the present invention.

Claims

1. A radio frequency fingerprint recognition method, characterized in that: Including: Step 1: M different devices transmit modulated signals. After the receiver receives the signals, a constellation trajectory diagram is obtained through processing; Step 2: Cluster the constellation points in the output constellation trajectory diagram of Step 1, calculate the within-class distance sum and the within-class average distance vector, and use the obtained clustering center points as RF fingerprints to construct a feature vector set; Step 3: Select A groups of feature vectors from the feature vector set as training samples for classification and B groups as test samples for classification, and respectively form a training set and a test set with the training samples and the test samples; Step 4: Use the training set for training to obtain trained classification model parameters; Among them, the classification model parameters include a weight vector and a separation margin; and the normal vector of the separation hyperplane in the weight vector feature space is denoted as w; the separation margin is the distance between the separation plane and the coordinate origin, denoted as ρ; Step 4, the specific process includes the following sub-steps: Step 4.1: Perform random Fourier feature transformation on the training samples in the training set to obtain a feature vector data set Z after random Fourier feature transformation; Step 4.2: Randomly select a data point from the feature vector data set Z after random Fourier feature transformation as the initial weight vector w; Step 4.3: Calculate the separation margin ρ based on w through golden section line search; Step 4.

4. Average all data points z(x j ) for which wz(x j ) - ρ < 0 to obtain the weight vector w m ; where x j is the data of the training set, and z(x j ) is the corresponding item of x j in the feature vector dataset Z; Step 4.5: Calculate the new weight vectors w and w through the stochastic gradient descent method m ; then use w new to update the weight vector w; new ​ Among them, use w new to update the weight vector w, that is, let w = w new ; Step 4.6: Repeat steps 4.3 to 4.5 until w basically stabilizes after K iterations, and denote w as the optimal weight vector w * , and calculate the optimal weight vector w according to the golden section line search * The corresponding optimal separation margin ρ * ; Among them, the optimal weight vector w * and the optimal separation margin ρ * are the trained model parameters; Step 5: Classify the test set, specifically including the following sub-steps: Step 5.1: Perform random Fourier feature transformation on the test set, obtain the random Fourier feature transformation according to formula (1), and obtain the test feature vector data set Z t ; Step 5.

2. Calculate the result of w * , ρ * w * z t (x ti ) - ρ * , denoted as P; Among them, the dimension of P is 1×n t ; n t is the number of elements in the test set; x ti is the i-th element in the test set, z t (x ti ) is an element in the test feature vector dataset Z ti after the random Fourier feature transformation of x t ; Step 5.3: Make a judgment according to the element values of P obtained in Step 5.2, specifically: if P(v)≥0, then the data to be classified corresponding to the v-th element in the test set is of the same class as the training set data; otherwise, if P(v)<0, then the data to be classified corresponding to the v-th element in the test set is of a different class from the training set data.

2. The radio frequency fingerprint recognition method according to claim 1, characterized in that: Step 1, specifically includes the following sub-steps: Step 1.1: Collect N segments of samples for each segment of the signal, sample at a sampling rate that satisfies the Nyquist law, and normalize the energy of the sampled signal; Step 1.2: Use a delay device to control the delays of the I-channel and Q-channel signals respectively for the I-channel and Q-channel signals in the energy-normalized signal output in Step 1.1, and perform differential processing at a certain interval; Step 1.3: Directly plot the differentially processed signal output in Step 1.2 in the space with the I-channel and Q-channel as the coordinate axes to obtain the corresponding constellation trajectory diagram.

3. The radio frequency fingerprint recognition method according to claim 2, characterized in that: Step 2, specifically: Step 2.1: Cluster the constellation points in the constellation trajectory diagram output in Step 1 to obtain different clustering centers of the densely distributed points in the constellation trajectory diagram; Step 2.2: According to the different clustering centers of the densely distributed points in the constellation trajectory diagram obtained in Step 2.1, calculate the within-class distance sum and the within-class average distance vector of each cluster; Step 2.3: Sort the calculated within-class distance sums and within-class average distance vectors of each cluster in a clockwise order according to the phase magnitudes of the clustering center points of each cluster to obtain the sorted within-class distance sums and within-class average distance vectors of each cluster, and form a feature vector set; Step 2.4: Traverse different signal-to-noise ratios and M devices, and repeat Steps 1 to 2.3 to obtain feature vector sets extracted from signals of different devices under different signal-to-noise ratios.

4. The radio frequency fingerprint identification method according to claim 3, characterized in that: In step 4.1, the calculation formula for the random Fourier feature transform is Equation (1): where x i is the data of the training set, with the feature dimension being d and the number of data being n; P(·) is the Gaussian distribution, σ is the kernel bandwidth, h u (d×1) is a randomly generated matrix; u = 1, …, d RF ; d RF is the dimension of the random Fourier feature transform, z(x i ) is the corresponding item of x i in the feature set Z.

5. The radio frequency fingerprint recognition method according to claim 4, characterized in that: In step 4.6, the value range of K is greater than 15 and less than 50.

6. The radio frequency fingerprint recognition method according to claim 5, characterized in that: In step 4.6, the condition for basically stabilizing is that the modulus difference of w is less than 0.0001.

7. A radio frequency fingerprint recognition method according to claim 6, characterized in that: In step 5.3, the value range of v is from 1 to n t .

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